Basis expansions in applied mathematics

Basis expansions are an extremely useful tool in applied mathematics. By using them, we can express a function representing a physical quantity as a linear combination of simpler ``modules'' with well-known properties. They are particularly useful for the applications described in this thesis. Perhaps the best known expansion of this type is the Fourier series of a periodic function, as decomposition into the infinite sum of simple sinusoidal and cosinusoidal elements, originally proposed by Fourier to study heat transfer. This dissertation employs some mathematical tools on problems taken from various areas of Engineering, exploiting their expansion properties: 1) Non-integer bases, which are applied to mathematical models in Robotics (Chapter 2). In this Chapter we study, in particular, a model for snake-like robots based on the Fibonacci sequence. It includes an investigation of the reachableworkspace, a more general analysis of the control system underlying the model, its reachability and local controllability properties. 2) Orthonormal bases, Riesz bases: exponential and cardinal Riesz basis with perturbations (Chapter 3). In this Chapter we obtain also a stability result for cardinal Riesz basis in the case of complex perturbations of the integers. We also consider a mathematical model for energy of the signal at the output of an ideal DAC, in presence of sampling clock jitter. When sampling clock jitter occurs, the energy of the signal at the output of ideal DAC does not satisfies a Parseval identity. Nevertheless, an estimation of the signal energy is here shown by a direct method involving cardinal series. 3) Orthogonal polynomials (Chapter 4). In this Chapter we introduce a new sequence of polynomials, which follow the same recursive rule of the well-known Lucas-Lehmer integer sequence. We show the most important properties of this sequence, relating them to the Chebyshev polynomials of the first and second kind. We discuss some relations between zeros of Lucas-Lehmer polynomials and Gray code. We study nested square roots of 2 applying a "binary code" that associates bits 0 and 1 to + and - signs in the nested form. This gives the possibility to obtain an ordering for the zeros of Lucas-Lehmer polynomials, which take the form of nested square roots of 2. These zeros are used to obtain two new formulas for Pi.